DFT, Rectangular/Ramp/Triangle/Cosine Pulse.

Cuthbert Nyack
The DFT and DTFT of a sequence of 100 samples is shown in the applet below. Samples between n1 and n2 is 1 and all others are zero. n1 and n2 changes the width of the pulse and Nf2 changes the width of the portion of the spectrum displayed. Changing n1 and n2 while leaving n2 - n1 constant translates the pulse.
The magnitude of the spectra has the characteristic sinc shape.
eg parameters (31, 36, NA, NA, NA, 0, 100, 50.0, 0.1, 1.0) show the spectra for 6 samples. Increasing n2 to 71 and reducing Vg to 0.15 show how the spectra changes as width of the pulse is increased.

The numerical values of the DFT and DTFT can be seen by changing wL.
Eg(31, 51, NA, NA, NA, 0, 45, 50.0, 0.1, 0.29) show the magnitude and phase of the DFT and DTFT at 1.88rad/s (0.03ws) is 9.752 and -82.8°.
Changing n1 to 32 and n2 to 52 leaves the magnitude unchanged at 9.752 but changes the phase to ~ -93.6° ie a change of (3 * 360/100)°.

Changing Fn to 1 and 2 shows the corresponding effects for the Ramp and Triangular pulse, 3 shows a 2 sided ramp, 4 shows 1 period of a cosine -p to +p halved and increased by 0.5, 5 shows a cosine from 0 to 2p halved and increased by 0.5, 6 shows a cosine from 0 to 2p.
Note the increasing concentration of energy in the main lobe as the pulse changes from Rectangular to Triangle to Cosine.

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COPYRIGHT 2007 Cuthbert Nyack.